# 数学代写|最优化理论作业代写optimization theory代考|QUASILINEARIZATION

#### Doug I. Jones

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## 数学代写|最优化理论作业代写optimization theory代考|QUASILINEARIZATION

Consider the two first-order, linear differential equations with timevarying coefficients
\begin{aligned} & \dot{x}(t)=a_{11}(t) x(t)+a_{12}(t) p(t)+e_1(t) \ & \dot{p}(t)=a_{21}(t) x(t)+a_{22}(t) p(t)+e_2(t) \end{aligned}
with specified boundary conditions $x\left(t_0\right)=x_0$ and $p\left(t_f\right)=p_f . a_{11}, a_{12}, a_{21}$, $a_{22}, e_1$, and $e_2$ are known functions of time, and $t_0, t_f, x_0$, and $p_f$ are known constants. It is desired to find a solution, $x^(t), p^(t), t \in\left[t_0, t_f\right]$, which satisfies the given boundary conditions. Notice that these differential equations are linear, but the boundary values are split.

First, suppose that we were to generate by numerical integration a solution, $x^H(t), p^H(t), t \in\left[t_0, t_f\right]$, of the homogeneous differential equations
\begin{aligned} & \dot{x}(t)=a_{11}(t) x(t)+a_{12}(t) p(t) \ & \dot{p}(t)=a_{21}(t) x(t)+a_{22}(t) p(t) \end{aligned}

with arbitrary assumed values for the initial conditions; as a convenient choice, let $x^H\left(t_0\right)=0$ and $p^H\left(t_0\right)=1$.

Next, we could determine a particular solution, $x^p(t), p^p(t)$, to the nonhomogeneous equations (6.4-1) by numerical integration, using $x^p\left(t_0\right)=x_0$ and $p^p\left(t_0\right)=0$ as initial conditions. Since the differential equations are linear, the principle of superposition applies, and
\begin{aligned} & x(t)=c_1 x^H(t)+x^p(t) \ & p(t)=c_1 p^H(t)+p^p(t) \end{aligned}
is a solution of (6.4-1) for any value of the constant $c_1$. We wish to find the solution that satisfies the specified boundary conditions. This can be accomplished by observing that
$$p\left(t_f\right)=p_f=c_1 p^H\left(t_f\right)+p^p\left(t_f\right) .$$
Solving this for $c_1$, which is the only unknown quantity, gives
$$c_1=\frac{p_f-p^p\left(t_f\right)}{p^H\left(t_f\right)}$$

## 数学代写|最优化理论作业代写optimization theory代考|Linearization of the Reduced State-Costate Equations

In general, the state and costate differential equations are nonlinear. Let us consider a simple situation in which there is one state equation and one costate equation. Assume that $\partial \mathscr{H} / \partial u=0$ has been solved for $u(t)$ and substituted in the state-costate equations to obtain the reduced differential equations
\begin{aligned} \dot{x}(t) & =a(x(t), p(t), t) \ \dot{p}(t) & =d(x(t), p(t), t) \end{aligned}

where $a$ and $d$ are nonlinear functions of $x(t), p(t)$, and $t$. Let $x^{(0)}(t), p^{(0)}(t)$, $t \in\left[t_0, t_f\right]$, be a known trajectory and $x^{(1)}(t), p^{(1)}(t), t \in\left[t_0, t_f\right]$, be any other trajectory. By performing a Taylor series expansion of the differential equations (6.4-13) about $x^{(0)}(t), p^{(0)}(t)$ and retaining only terms of up to first order, we obtain
\begin{aligned} \dot{x}^{(1)}(t) \doteq \dot{x}^{(0)}(t) & +\left[\frac{\partial a}{\partial x}\left(x^{(0)}(t), p^{(0)}(t), t\right)\right]\left[x^{(1)}(t)-x^{(0)}(t)\right] \ & +\left[\frac{\partial a}{\partial p}\left(x^{(0)}(t), p^{(0)}(t), t\right)\right]\left[p^{(1)}(t)-p^{(0)}(t)\right] \ \dot{p}^{(1)}(t) \doteq \dot{p}^{(0)}(t) & +\left[\frac{\partial d}{\partial x}\left(x^{(0)}(t), p^{(0)}(t), t\right)\right]\left[x^{(1)}(t)-x^{(0)}(t)\right] \ & +\left[\frac{\partial d}{\partial p}\left(x^{(0)}(t), p^{(0)}(t), t\right)\right]\left[p^{(1)}(t)-p^{(0)}(t)\right] \end{aligned}
or, substituting $a\left(x^{(0)}(t), p^{(0)}(t), t\right)$ for $\dot{x}^{(0)}(t)$ and $d\left(x^{(0)}(t), p^{(0)}(t), t\right)$ for $\dot{p}^{(0)}(t)$,
\begin{aligned} \dot{x}^{(1)}(t)= & a\left(x^{(0)}(t), p^{(0)}(t), t\right) \ & +\left[\frac{\partial a}{\partial x}\left(x^{(0)}(t), p^{(0)}(t), t\right)\right]\left[x^{(1)}(t)-x^{(0)}(t)\right] \ & +\left[\frac{\partial a}{\partial p}\left(x^{(0)}(t), p^{(0)}(t), t\right)\right]\left[p^{(1)}(t)-p^{(0)}(t)\right] \ \dot{p}^{(1)}(t)= & d\left(x^{(0)}(t), p^{(0)}(t), t\right) \ & +\left[\frac{\partial d}{\partial x}\left(x^{(0)}(t), p^{(0)}(t), t\right)\right]\left[x^{(1)}(t)-x^{(0)}(t)\right] \ & +\left[\frac{\partial d}{\partial p}\left(x^{(0)}(t), p^{(0)}(t), t\right)\right]\left[p^{(1)}(t)-p^{(0)}(t)\right] . \end{aligned}

# 最优化代写

## 数学代写|最优化理论作业代写optimization theory代考|QUASILINEARIZATION

\begin{aligned} & \dot{x}(t)=a_{11}(t) x(t)+a_{12}(t) p(t)+e_1(t) \ & \dot{p}(t)=a_{21}(t) x(t)+a_{22}(t) p(t)+e_2(t) \end{aligned}

\begin{aligned} & \dot{x}(t)=a_{11}(t) x(t)+a_{12}(t) p(t) \ & \dot{p}(t)=a_{21}(t) x(t)+a_{22}(t) p(t) \end{aligned}

\begin{aligned} & x(t)=c_1 x^H(t)+x^p(t) \ & p(t)=c_1 p^H(t)+p^p(t) \end{aligned}

$$p\left(t_f\right)=p_f=c_1 p^H\left(t_f\right)+p^p\left(t_f\right) .$$

$$c_1=\frac{p_f-p^p\left(t_f\right)}{p^H\left(t_f\right)}$$

## 数学代写|最优化理论作业代写optimization theory代考|Linearization of the Reduced State-Costate Equations

\begin{aligned} \dot{x}(t) & =a(x(t), p(t), t) \ \dot{p}(t) & =d(x(t), p(t), t) \end{aligned}

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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